Very frequently in SRT, for "simplifying" the description of collisions,
the technique of transition to any "conveniently moving" frame of reference
is used. Such a procedure, however, has no physical grounds, and the
principle of relativity for closed identical systems is for
nothing here at all. If the relativistic experiments are carried out on
artificial beams of particles, then the sources (accelerators) and recording
instruments are bound to the Earth, and accelerators and instruments will
not fly, together with a moving observer, from our mental imagination only.
If some process in Wilson's chamber is investigated, then the tracks of
particles are bound to a medium (that is, to Wilson's chamber), rather than
to a flying observer. For example, in the classical physics the angle between
the tracks of particles will not change due to motion of an observer.
At the same time, the angle between the velocities of particles, which leave
mentioned tracks, can depend on observer's motion velocity. In the
relativistic physics the angles between trajectories and between velocities
of particles depend, also according to various laws, on observer's motion
velocity. Therefore, such a seemingly probable from SRT viewpoint transition to a new frame
of reference can essentially distort the interpretation of a solution.
That is, any process should be considered in the frame of reference of a real
observer (or recording instrument) only.

One more distortion of reality is the consideration of the process of
collision of two particles (being principally point-like in SRT) as a planar
motion. In fact, to fit to an ideal problem of two points, a measuring device cannot
simultaneously fly with each pair of particles and differently rotate even in
studies of statistical characteristics of point particles: the position of the
device is fixed. Besides the point-like particles should be considered as a limiting
case of particles having real finite size (otherwise no frontal collisions
would be observed, it would be impossible to consider collisions of atoms
and molecules, the protons would not have structure, etc.). And in the
present case the collision of particles is principally three-dimensional
(the probability of planar motion is zero). Let, for example, two identical
balls (1 and 2) to approach each other before collision over straight
lines crossing in space (the minimum distance between skew straight lines is
smaller than the ball diameter). Even from the very beginning of the
experiment we cannot draw the plane through these specified straight lines.
Nevertheless, we shall take the middle of a minimum distance between crossing
straight lines (the trajectories before collision) and draw through it
intersecting straight lines parallel to the given trajectories.

Figure 4.8:
Nonplanar motion of two particles.

Now, only one plane passes through intersecting straight lines
(Fig. 4.8). The centers of balls move parallel to this plane before collision:
the first ball's center moves slightly above the plane and the second ball's
center - slightly below this plane. After collision the balls will fly over
other crossing straight (skew) lines. And, again, it is impossible to draw the
plane through these skew straight lines. Again, we shall perform a similar
procedure with parallel transition of straight lines, on which the lines of
motion lie after collision, before intersecting at the middle. We shall draw
through intersecting straight lines the plane (the centers of balls will
again move on different sides from this plane). However, "the plane before
collision" does not coincide with "the plane after collision", but intersects
it at some angle.

Second method: let us draw one plane through the trajectory of motion
of the first particle (intersecting straight lines of its motion before and
after collision), and the second plane - through a similar
trajectory of motion of the second particle. However, these planes are also
intersected at some angle (Fig. 4.9).

Figure 4.9:
Three-dimensionality of collision of two particles.

So, what follows from three-dimensionality of motion? First. Not all
relations turn out linear ones. For example, the distance
between bodies occurs some nonlinear function of time even for the rectilinear
uniform motion of the bodies along skew lines. Second.
We shall write the classical laws of conservation of momentum (in projections)
and of energy:

(4.3)

(4.4)

(4.5)

(4.6)

We see from (4.3-4.6), that for six unknown quantities
() there are four
equations only. Thus, there should remain two
indefinite parameters in the solution. If we suppose the motion to be planar
(i.e. exclude equation (4.5)), then for remaining four unknowns we shall have
three equations. Therefore, in comparing SRT results with the classical physics
the substitution of solutions is accomplished, and there remains only
one indefinite parameter
(the scattering angle is usually considered to be the latter one).
Such a substitution results in improper interpretation of the experimental
data, especially when the missed quantities are restored. For example, the
book [33] demonstrates two tracks of fly-away of particles of identical mass
and charge (more correctly, of identical ratio) with dispersion angle
lower than , and the conclusion on the classical mechanics
invalidity is drawn from this demonstration. Let us write the expression
for angle between the trajectories of dispersed particles:

(4.7)

Choose axis so, that it will be . Now we express
variable from equation (4.3), variable - from equation (4.4),
variable - from equation (4.5), and from equation (4.6) we shall
express quantity (in this case the condition
restricts the region of possible values of all variables). Substitute all
aforementioned quantities into equation (4.7). As a result, we obtain the
two-parametric dependence on and , which is not written
here because of its awkwardness. Using graphical programs, we can be
convinced that for the given values of
we obtain some surface similar to the inner part of a cylinder; that is,
quantity varies within wide limits. For example, it can
easily be verified that the values

satisfy all classical conservation laws (4.3-4.6). For these values we obtain
, that is, . Note: if the
velocities are assumed to be expressed in terms of the speed of light,
then a lower velocity is quite real for the motion of internal electrons in
atoms beginning with . And, generally, nobody saw electrons in
atoms being at rest! The angle of is unambiguously obtained in
the classical physics at collision with a particle being at rest in the
coordinate system of a recorder (but only where such a particle can be found?).
However, the observed fly-away angle of does not result at all
in an unambiguous opposite assertion, that one of particles had been at rest
(the mathematical probability of such an event is infinitesimal). Thus, the
reverse problem of
restoring the missed data is not an unambiguous procedure either in the
classical, or in the relativistic physics (there exists an infinite number
of various self-consistent solutions).

For more rigorous verification of conservation laws in collisions (independent
on any theory) it is necessary to study collisions of particles in vacuum for narrow
monoenergetic beams of known particles for the given collision angles. In this
case the complete study of the collision process should include the check of the
energy balance of particles (for each scattering angle in space), the testing
the balance of momenta of particles, the testing the balance of the total
number of particles in beams before and after collision (the probability of
scattering), the control of the balance of arising
radiation in energies and directions. There are two more questions (two more
uncertainties), which are not usually emphasized, namely: does the scattering
depend on a mutual orientation of spins of colliding particles? And do these
spins change during the collision? In the classical physics the answer
to these questions is "yes" (but in the quantitative respect it strongly
depends on the "structure" of balls).

The author did not meet any complete analysis of any collision process in
SRT with respect to all issues set forth above. This does not imply,
however, an unambiguous conclusion on invalidity (within the limits of
experimental errors) of usually utilized relativistic conservation laws
in any collision process (though this can quite occur to be the fact for
many separate cases). The author only asserts that there are no even
separate examples of absolute confirmation of relativistic collision laws
(to say nothing of the global confirmation).

From a principally rigorous position, the application of relativistic
conservation laws to the collision process in the elementary particle physics
is rather doubtful. Whether these laws can retain their form irrespective of
the charge of colliding particles, collision angles and dispersion angles?
You see, the charged particles undergo acceleration during the collision.
Therefore, in the modern concepts (in the SRT as well), some radiation (field)
should always be observed. Is it necessary,
really, to behave as the students having peeped at the answer to the problem:
if the instrument has recorded a -quantum ("has seized our hand"), then
it should be clearly taken into account "with a clever air". And should one
trust in validity of SRT formulas "with a clever air" in remaining cases
as well? So, where is the "predictive force" of SRT? Actually, the conservation
laws should be explicitly supplemented by the terms, which take into account
the energy and momentum of the field.

Generally speaking, the only case, where the discussion of relativistic
conservation laws at "collisions" is lawful, is the interaction of particles
with the forces of electromagnetic nature (the Lorentz force). For remaining
cases the fulfillment of relativistic conservation laws is an unverified
hypothesis (the light spheres of SRT bear no relation to the forces of
non-electromagnetic nature). However, in the case of electromagnetic
interactions no SRT ideas are required for deriving relativistic conservation
laws as well. It is known that the equations of motion with the initial
conditions completely determine all characteristics of motion, including the
integrals of motion. Such an integral of motion can be the energy (but not
always!). It follows from the equation of motion, that

(4.8)

Introduce the definition of the potential energy

Knowing the form of the momentum (this is a quantity appeared in the
experimental equation of motion (4.8); for example, in the classical case
and in the relativistic case
), one can obtain the energy
conservation law from :
classical , or relativistic
, respectively. Under the condition
of equality forces of action and counteraction (the third Newton's law, the
hypothesis of central forces) we have: .
Then from the equation of motion (4.8) we can obtain the momentum conservation
law (this is again a quantity appeared in the experimental equation of
motion (4.8)): from
we obtain

However, in the presence of magnetic forces ,
and the relativistic law of conservation of momentum of particles can be
violated in the general case. Since the majority of particles, even many
electrically neutral ones, have magnetic moment (i.e. they represent not
"ideal point charges of the SRT", but charged
magnetic rotators of finite size), the application of the relativistic
momentum conservation law in the nuclear physics and elementary particle
physics without explicit considering the field momentum is completely
illegitimate. Therefore, we again arrive at the necessity of explicit
considering the momentum (and, hence, the energy) of the field at collisions.
(Possibly, this will help to regulate the nuclear physics and elementary
particle physics and to decrease the number of particles-ghosts?)

The account taken of the radiation reaction force also results in violation
of energy and momentum conservation laws declared in SRT. Should we refuse
from accounting this force in the process of collision of particles? But
this force just should be most significant in this process (there are great
fields, owing to rapprochement of high-energy particles, and great
variable accelerations).